Optimal. Leaf size=105 \[ \frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {32 a^3 \left (a x^2+b x^3\right )^{3/2}}{315 b^4 x^3}+\frac {16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x} \]
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Rubi [A]
time = 0.08, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027,
2039} \begin {gather*} -\frac {32 a^3 \left (a x^2+b x^3\right )^{3/2}}{315 b^4 x^3}+\frac {16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2027
Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int x^2 \sqrt {a x^2+b x^3} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {(2 a) \int x \sqrt {a x^2+b x^3} \, dx}{3 b}\\ &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}+\frac {\left (8 a^2\right ) \int \sqrt {a x^2+b x^3} \, dx}{21 b^2}\\ &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}+\frac {16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}-\frac {\left (16 a^3\right ) \int \frac {\sqrt {a x^2+b x^3}}{x} \, dx}{105 b^3}\\ &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {32 a^3 \left (a x^2+b x^3\right )^{3/2}}{315 b^4 x^3}+\frac {16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.50 \begin {gather*} \frac {2 \left (x^2 (a+b x)\right )^{3/2} \left (-16 a^3+24 a^2 b x-30 a b^2 x^2+35 b^3 x^3\right )}{315 b^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 57, normalized size = 0.54
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-35 b^{3} x^{3}+30 a \,b^{2} x^{2}-24 a^{2} b x +16 a^{3}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{315 b^{4} x}\) | \(57\) |
default | \(-\frac {2 \left (b x +a \right ) \left (-35 b^{3} x^{3}+30 a \,b^{2} x^{2}-24 a^{2} b x +16 a^{3}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{315 b^{4} x}\) | \(57\) |
risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-35 b^{4} x^{4}-5 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}-8 a^{3} b x +16 a^{4}\right )}{315 x \,b^{4}}\) | \(61\) |
trager | \(-\frac {2 \left (-35 b^{4} x^{4}-5 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}-8 a^{3} b x +16 a^{4}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{315 b^{4} x}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 53, normalized size = 0.50 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 62, normalized size = 0.59 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{315 \, b^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {x^{2} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 131, normalized size = 1.25 \begin {gather*} \frac {32 \, a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )}{315 \, b^{4}} + \frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} \mathrm {sgn}\left (x\right )}{b^{3}}\right )}}{315 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.49, size = 62, normalized size = 0.59 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}\,\left (-16\,a^4+8\,a^3\,b\,x-6\,a^2\,b^2\,x^2+5\,a\,b^3\,x^3+35\,b^4\,x^4\right )}{315\,b^4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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